DOI :
10.2240/azojomo0268
Jul 10 2008
C. Corvalán Moya, M. J. Iribarren and F. Dyment
Copyright ADTECH; licensee AZoM.com Pty Ltd.
This is an AZo Open Access Rewards System (AZoOARS) article distributed under the terms of the AZo–OARS https://www.azom.com/oars.asp which permits unrestricted use provided the original work is properly cited but is limited to noncommercial distribution and reproduction.
AZojomo (ISSN 1833122X) Volume 4 July 2008
Topics Covered
Abstract
Introduction
Experimental Procedure
Results and Discussion
Kinetic Type B
Kinetic Type C
Discussion
Conclusions
Acknowledgements
References
Contact Details
Abstract
Atomic transport along grain boundaries (GB) and interphase boundaries (IB) plays a key role in a large number of metallurgical processes.
The experimental determination of GB or IB diffusion coefficients is very similar to that for the corresponding bulk diffusion, the main difference being the relation between penetration and diffuser concentration. In this framework, Harrison’s classification allows us to deal with different kinetics (type A, B and C). The proper employment of type B and C kinetics in different ranges of temperature provides the opportunity to obtain not only extended diffusion parameters, but additional information about the physical factors included in the “apparent diffusion coefficient” in grain boundaries (P = s D_{gb}).
In this paper, diffusion parameters of cobalt along the grain boundaries of pure αZr are presented. In order to cover a range of temperature of technological interest, the interval 430633 K was studied. Suzuoka and Gaussian solutions for type B and C kinetics were employed. Also, diffusion parameters of cobalt along grain boundaries of ßZr 20% Nb are presented in a temperature interval of 710875 K, according to B kinetics.
The results show that cobalt behaves as an ultrafast diffuser in αZr and ßZr 20% Nb grain boundaries. In αZr, comparison between B and C kinetics allows us to estimate the segregation factor, its temperature dependence and hence, to obtain the true value of D_{gb} even in the range of temperature where only P is available.
Keywords
Diffusion, Pure Zr, ZrNb Alloys, Grain Boundaries, Segregation
Introduction
Atomic transport along both grain boundaries (GB) and interphase boundaries (IB) is orders of magnitude faster than in the bulk crystal. For this reason both GB and BI play a key role in a large number of metallurgical processes such as plastic deformation, corrosion at high temperatures, solidstate transformations, surface treatments and the stability of precipitates in a matrix.
Zrbased alloys (mainly Zircaloys and Zr 2.5% Nb) are extensively used in nuclear and chemical industries due to their excellent corrosion resistance, mechanical properties and nuclear properties. They are usually employed in polycrystalline forms, exhibiting a high density of GB and IB.
In this context, solute and self diffusion studies in the bulk and in GB and IB are relevant from a basic and an applied point of view. In most nuclear power installations, Fe, Co, Ni and Cr are normally present as solutes or impurities in devices which are in contact with Zr. For instance, different types of steels, including stainless steels, are used along with Zircaloys or ZrNb alloys. Elements such as Fe, Co, Ni and Cr show diffusion coefficients several orders of magnitude faster than selfdiffusion in Zr and they are usually referred as ultrafast diffusors.
The experimental determination of the diffusion coefficients in GB and IB is similar to that for bulk diffusion, except that the relation between the concentration of the diffusor and the penetration is approximately linear instead of parabolic.
Experimental Procedure
The material was cut into samples around 0.75 cm2 surface area. They were mechanically polished finishing with 1 micron diamond paste. Some of the samples were employed for metallographic purposes in order to correlate the real morphology with the mathematical models, for the correct application of the different possible kinetics. Zr20%Nb samples were annealed at 1223 K for 2 hours, and then quenched, obtaining a stabilized grain boundary network structure, while pure αZr samples showed a well stabilized structure of small grain size.
The radiotracer 60Co was obtained in the laboratories of the National Atomic Energy Comission of Argentina.
Cobalt powder was irradiated in the Nuclear Reactor RA1; the irradiation time was calculated in order to obtain the minimun activity suitable for our experiments. For the dissolution of the Co two different reagents were assessed: nitric and hydrochloric acids. As a first attempt, non irradiated Co was used. According to this experience, nitric acid was chosen since it generates a minor amount of active solution and the dissolution is completed in a shorter time. In addition, according to data obtained from the bibliography, the use of nitric acid in preference to hydrochloric acid diminishes the damage caused by corrosion of the samples, which, if it happens, would disturb the diffusion results. The 60Co was obtained, therefore, in a HNO_{3} solution and it was deposited directly onto the polished surface of the samples in order to make a diffusion couple. The samples were wrapped in high purity tantalum foil and vacuum sealed in quartz tubes with a slight overpressure of high purity argon. The diffusion anneals were performed in Adamel electrical furnaces, provided with P+I+D electronic control, with a precision of ± 1 K.
Later, the samples were turned radially, to assure a flat front diffusion from the deposited face. The sectioning of the samples was performed in a precision abrasion machine. The thicknesses of the removed layers were determined by weight difference of the samples after each sectioning. The measurements were made with a 10^{5}g accuracy electronic balance; the density of the material involved and the area of the samples were known.
An INa(Tl) well detector was employed in order to measure the activity (concentration) of the layers, this information being collected and analyzed by means of a multichannel analyzer device.
The quoted errors were estimated around 15% for the grain boundary diffusion parameters P_{gb} and D_{gb}.
Results and Discussion
Diffusion in grain boundaries is a complex process that includes the serial elemental processes:
 bulk diffusion from the surface of the sample
 diffusion along the GB
 bulk diffusion from the GB
According to the relative importance of these processes, different situations or kinetics could result. Harrison [1] allows us to deal with three different kinetics: types A, B and C. We will give the expressions corresponding to B and C kinetics, both used in this work.
Kinetic Type B
Main processes: diffusion along GB and bulk diffusion from GB.
100 d < bulk diffusion distance = (Dt)^{1/2} < d/20
where d is the mean grain size and a is the GB thickness.Then we can apply Fisher’s model [2] of an isolated GB.
In this framework, and in order to obtain the penetration profile, the following equations are applied:
Constant source
^{} (1)
Instantaneous source
(2)
where s is the segregation factor; D is the bulk diffusion coefficient and P_{gb} = δ D_{gb} is the experimentally measurable parameter, called “apparent diffusion coefficient“. The B kinetic is the more employed one because it allows reasonable times and temperatures for the diffusion anneals.
Figure 1. Schematic illustration of diffusion B kinetic.
Kinetic Type C
Main process: diffusion along GB. The bulk diffusion from the surface and from the GB is meaningless.
The penetration profile in this kinetic, equivalent to bulk diffusion, is either gaussian (instantaneous source) or an error function (constant source). D_{gb} is directly measured. For a Gaussian solution
C(x, t) = Co exp (x^{2}/4D_{gb}t) (3)
where Co is the initial diffusor concentration.
Figure 2. Schematic illustration of diffusion C kinetic.
Figure 3 shows a typical penetration profile in B kinetic obtained in αZr GB.
Figure 3. GB Diffusion in αZr. B Kinetic.
The temperatures and annealing times of the experiments done with this kinetic were determined using data from Vieregge and Herzig [3]. The range of temperature used in this work was very low and the maximum mean square bulk diffusion distance calculated was 0.4 µm. As the grain size was between 7 and 10 µm, it was possible to consider that we have isolated grain boundaries, necessary for the B kinetic. Experiments at the same temperature for different times were considered unnecessary as the concentration profiles followed well the model used. It is important to remark that the structure of the αZr samples remains unchanged during the diffusion annealing. This is so because a possible change, for instance grain size, would be related to Zr bulk selfdiffusion. According to previous work, Co [4] is a diffusor about nine orders of magnitude faster than Zr selfdiffusion [5] in the bulk. As can be seen in [3], this is also similar for GB diffusion: Co diffusion along αZr GB is eight orders of magnitude faster than Zr GB selfdiffusion in the range of temperature involved in that work. When estimating the maximum segregation of Co in αZr, the authors of [3] considered that about four decades of that difference have to be ascribed to different GB diffusion coefficients.
In the case of kinetic type B, P_{gb} are the values experimentally measured; these values are given in Table 1.
Table 1. "Apparent diffusion coefficients" in αZrZr GB
Figure 4 shows a typical penetration profile in C kinetic obtained in αZr GB.
Figure 4. GB Diffusion in αZr. C Kinetic.
In the case of kinetic type C, the values experimentally measured are D_{gb} directly. The importance of this kinetics is that we do not need to know the values of the thickness of the GB and the segregation factor. The measured values in this work are shown in Table 2.
Table 2. Diffusion coeficients in αZr GB
In Figure 5, the Arrhenius plot shows previous experimental data [3] and results from this work: measurements in B kinetic, P_{gb}, and the product a. D_{gb} in C kinetic. It is possible to see that the influence of the segregation factor is more important at lower temperatures.
Figure 5. Arrhenius diagram of Co diffusion along αZr GB. B and C kinetics.
For the diffusion in Zr20%Nb, Figure 6 shows a typical penetration profile in kinetic type B and Table 3 gives the P_{gb} values measured.
Figure 6.GB Diffusion in Zr20%Nb. B Kinetic
Table 3. Apparent diffusion coefficients“ in ßZr 20% Nb GB
T/K 
t/s 
P_{gb}/ m^{3}s^{1} 















Figure 7. Arrhenius diagram of Co diffusion along ßZr20%Nb GB.
Discussion
This work presents results of the diffusion of Co in GB of pure αZr in the range of temperatures 430633 K and the diffusion of Co in GB of ßZr 20% Nb in the range of temperatures 710875 K. In αZr this is the first time where the kinetic type C has been applied.
The results show that Co is an ultrafast diffuser in GB of pure αZr and ßZr 20% Nb as it is in bulk αZr. In both cases the relation between GB and bulk diffusion [5, 6] is around 10^{6}, in agreement with the theory of diffusion by shortcircuit paths. In both materials the Arrhenius plot is straight, indicating the operation of a unique diffusion mechanism.
In αZr, the measured values of P_{gb} in type B kinetic show a good coincidence with the values measured in [3]. The concentration of impurities, particularly Fe, is very similar in both materials (192 ppm of Fe in [3], 138 ppm in our case), the main difference being the O content, which is larger in our case.
The comparison between B and C kinetics lets us obtain a direct evaluation of the segregation factor s of Co in the αZr GB, according to equation (4)
(4)
For calculating s as the ratio between P_{gb} and d D_{gb} we need to take into account Vieregge and Herzig´s work at the lower temperatures; then it is necessary to consider the analysis made by the authors with respect to those values. To explain the lower activation energy obtained below 600 K, they considered the possibility of the coexistence of different diffusion paths in grain boundaries and then, the dominance of those processes with lowest activation energies. They also considered the possibility of strong segregation of an impurity such as Fe with the resultant formation of new phases in the grain boundaries that could lead to a deviation from the Henrytype segregation behaviour. In [3], the Fe content of the material used for the lower temperatures is really very low (20 ppm) as compared to the one used at higher temperatures (192 ppm). Finally they mentioned the possibility of using too large values of Co bulk diffusion, D, in the expressions used to calculate P_{Co}. To obtain a straight line in the whole temperature range covered in [3] would imply a strong downward curvature of the Arrhenius plot of bulk diffusion of Co in αZr, as the square root of D appears in the expressions (1) or (2). In this work, we do not neglect the possibility of a curvature. In fact, in the work by Pérez, Nakajima and Dyment [7] this characteristic is carefully analyzed. Nevertheless, all the positive curvature of Co P_{gb} in αZr obtained in [3] would not be cancelled with the negative curvature of Co bulk diffusion in αZr discussed in [7]. Taking into account the former arguments and the fact that we have measured using both kinetics, a first approximation to the segregation coefficient value can be made.
The values of s at the different temperatures were calculated assuming a = 5x10^{10} m. They are given in Table 4. “s” shows an important variation with the temperature, from roughly 50 to 15000 over a range of 50 K, as can be seen in Figure 8.
Table 4. Segregation factor of Co along αZr GB.
T/K 
P_{gb}/ m^{3}s^{1} 
D_{gb}/ m^{2}s^{1} 
S 
430 
4,5x10^{24} 
6x10^{19} 
15000±4500 
440 
6x10^{24} 
1x10^{17} 
1200±360 
460 
1x10^{23} 
8,7x10^{17} 
230±69 
487 
6x10^{23} 
2,6x10^{15} 
46±14 
Figure 8. Temperaturedependence of the GB segregation factor, s , of Co in αZr.
Considering the possible influence of Fe on Co diffusion and its different contents between [3] and this work, we think that a fraction of the ratio P_{gb} / d D_{gb} could not be attributed to the pure segregation factor. A small uncertainty of the s value could be assumed, taking into account that at the low temperatures Vieregge and Herzig used a purer Zr with only 20 ppm of Fe. This point deserves further investigation, and we are working in this direction.
Figure 9 shows the Arrhenius plot of Co GB diffusion in: pure αZr (Vieregge and Herzig´s work [3] and the present work) and in ßZr 20% Nb (present work); and for comparison, in α/ß IB diffusion in Zr 2.5% Nb [6]. First of all there is observed a reduction of about 4 orders of magnitude between GB diffusion in pure αZr and the diffusivity in GB of Zr20%Nb and in IB of Zr2.5%Nb. Also, it is possible to see a slight reduction of the diffusivity with the increase of Nb between IB diffusion in Zr2.5%Nb and GB diffusion in Zr20%Nb; this difference remains within one order of magnitude in the range of temperature measured. It is necessary to have in mind that two different kinds of short circuit paths are involved: pure ß GB in the present work and α/ß IB in [6]. A question arises: is the difference obtained related to a different diffusion mechanism along a grain boundary as compared to the other short circuit path, or is the segregation factor mainly responsible for the difference?
Figure 9. Arrhenius plot of Co GB diffusion along pure αZr, ßZr 20% Nb and a/ß diffusion in Zr 2.5% Nb.
In [6] for Co and in [8] for Cr it was interpreted that in IB diffusion in Zr 2.5% Nb mostly the aphase acts in the complex mechanism involved in these short circuits, due to the very limited solubility of Co and Cr in the aphase. If this is the case, it seems that for Co diffusion, the amount of Nb present in the alloy is a second order effect compared to the fact of the existence of the ß phase. This subject deserves more data to be elucidated and we are working in this direction.
Related to the difference found between P_{gb} in pure αZr compared to P_{gb} in Zr20%Nb and P_{ib} in Zr2.5%Nb (based on Figure 9 as well as Table IV), the segregation factor could only explain between one and two orders of magnitude in the P´s difference in the region of high temperatures. So, D_{gb} in αZr could be at least two orders of magnitude higher than D_{gb} (in αZr20%Nb) and D_{ib} (in αZr/ß Zr2.5%Nb) respectively.
Conclusions
This paper presents results of the diffusion of Co in GB of pure a Zr in the temperature range 430633 K and the diffusion of Co in GB of ß Zr 20% Nb in the temperature range 710875 K. In αZr these are the first results where the kinetic type C has been applied.
The results show that Co is an ultrafast diffuser in GB of pure αZr and in ßZr 20% Nb as it is in bulk αZr. In both materials the Arrhenius plot is straight, indicating the operation of a unique diffusion mechanism.
In αZr, the results obtained for P_{gb} in kinetic type B show a good coincidence with the values measured in [3]. This fact and the comparison between B and C kinetics lets us to obtain a direct evaluation of the segregation factor s of Co in the αZr GB; extending this estimation towards higher temperatures, where only P_{gb} is available, the extrapolation of D_{gb} values (type C kinetics) seems to be applicable. Therefore, Co D_{gb} in αZr seems to be at least two orders of magnitude higher than D_{gb} in ßZr20%Nb and D_{ib} in αZr/ß Zr2.5%Nb respectively.
Acknowledgements
The authors wish to thank the support of CONICET (PIP n° 5322) and Agencia Nacional de Promoción Científica y Tecnológica (PICT 20479) for the grants obtained.
References
1. G. Harrison, “Influence of Dislocations on Diffusion Kinetics in Solids with Particular Reference to the Alkali Halides”, Trans. Faraday Soc., 57 (1961) 11911199.
2. J.C. Fisher, “Calculation of Diffusion Penetration Curves for Surface and Grainboundary Diffusion”, J. Appl. Phys., 22 (1951) 7484.
3. K. Vieregge and Chr. Herzig, “Grain Boundary Diffusion in azirconium : Part II: Fast Diffusing Cobalt Bulk Interstitials“, J. of Nucl. Mat., 175 (1990) 2941.
4. G.V. Kidson, “The Diffusion of /sup 58/Co in Oriented Single Crystals of Alpha –zirconium“, Philos. Mag. A, 44 (1981) 341355.
5. J. Horvath, F. Dyment and H. Mehrer, “Anomalous Selfdiffusion in a Single Crystal of azirconium“, J. Nucl. Mat., 126 (1984) 206214.
6. O.E. Agüero, M.J. Iribarren and F. Dyment, “CoDiffusion Along the Alpha/Beta Interphase Boundaries of Zr2.5%Nb Alloy”, Defect and Diffusion Forum, 194199 (2001) 12111216.
7. R.A. Pérez, H. Nakajima and F. Dyment, “Diffusion in aTi and Zr: Diffusion in Materials and its Application: Recent Development”, Materials Transactions JIM, 44 (2003) 213.
8. M. J. Iribarren, M. M. Iglesias and F. Dyment, “Diffusion along Grain and Interphase Boundaries in Alpha Zr and Zr2.5 Wt Pct Nb Alloy”, Met. and Mat. Trans. A, 33 (2002) 797800.
Contact Details
*C. Corvalán Moya and M. J. Iribarren
Comisión Nacional de Energía Atómica (CNEA) Av. del Libertador 8250 (1429) Buenos Aires República Argentina
Email: [email protected] 
C. Corvalán Moya and F. Dyment
Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina (CONICET) 
This paper was also published in “Advances in Technology of Materials and Materials Processing Journal, 9[2] (2007) 161166”.